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Those Fascinating Numbers
Page88(108 of 451)

88 Jean-Marie De Koninck 462 • the 12th number n such that n · 2n − 1 is prime (see the number 115); • the smallest number which cannot be written as the sum of eight non zero distinct squares (R.K. Guy [101], C20). 463 • the first term of the smallest sequence of five consecutive prime numbers all of the form 4n + 3 (as well as six or seven consecutive prime numbers all of the form 4n + 3); if we denote by qk the first term of the smallest sequence of k consecutive prime numbers all of the form 4n + 3, we have the following table: k qk 1 3 2 3 3 199 4 199 5 463 k qk 6 463 7 463 8 36 551 9 39 607 10 183 091 k qk 11 241 603 12 241 603 13 241 603 14 9 177 431 15 9 177 431 k qk 16 95 949 311 17 105 639 091 18 341 118 307 19 727 334 879 20 727 334 879 (see the number 2 593 for the similar question with 4n + 1); • the prime number which allows one to write the number 6 as the difference of two powerful numbers: 6 = 54 · 73 − 4632 = 214 375 − 214 369, a repre- sentation discovered by W. Narkiewicz and that S.W. Golomb thought to be impossible; Mollin & Walsh [141] proved that each integer k 0 has infinitely many representations as the difference of two powerful numbers. 464 • the third solution of σ(n) = 2n + 2: the list of solutions of this equation begins as follows: 20, 104, 464, 650, 1 952, 130 304, 522 752, 8 382 464,. . . 91 467 • the prime number which appears the most often as the tenth prime factor of an integer (see the number 199). 469 • the 15th number n such that n! − 1 is prime (see the number 166). 91It is easy to show that each number n = 2α·p, where α is a positive integer such that p = 2α+1−3 is prime, is a solution of σ(n) = 2n + 2: it is the case when α = 2, 3, 4, 5, 8, 9, 11, 13, 19, 21, 23, 28, 93; the solutions corresponding to α = 2, 3, 4, 5, 8, 9, 11 are included in the above list.